Gradient estimates for the insulated conductivity problem: the non-umbilical case

TL;DR

This paper extends gradient estimates for the insulated conductivity problem to general convex inclusions, revealing dependence on curvature and eigenvalues, and advancing understanding of solution behavior as inclusions approach each other.

Contribution

It provides new gradient estimates for convex inclusions in the insulated conductivity problem, incorporating curvature and spectral properties, generalizing previous results for special cases.

Findings

Gradient estimates depend on principal curvatures.

Estimates are characterized by the first non-zero eigenvalue.

Results apply to general convex inclusions, not just balls.

Abstract

We study the insulated conductivity problem with inclusions embedded in a bounded domain in $\mathbb R^n$, for $n \ge 3$. The gradient of solutions may blow up as $\varepsilon$, the distance between inclusions, approaches to $0$. We established in a recent paper optimal gradient estimates for a class of inclusions including balls. In this paper, we prove such gradient estimates for general strictly convex inclusions. Unlike the perfect conductivity problem, the estimates depend on the principal curvatures of the inclusions, and we show that these estimates are characterized by the first non-zero eigenvalue of a divergence form elliptic operator on $\mathbb S^{n-2}$.